Stanley's Lemma and Multiple Theta Functions
William Y. C. Chen and Lisa H. Sun
Abstract: We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions [(-1)^{δ}a_{1}^{α1}a_{2}^{α2}... a_{r}^{αr}q^{s} q^{t}]_{∞}, where α_{i} are integers, δ=0 or 1, s∈ Q, t∈ Q^{+}, and the exponent vectors (α_{1,}, α_{2},...,α_{r}) are linearly independent over Q. For an identity on such multiple theta functions, we provide an algorithmic approach for computing a system of contiguous relations satisfied by all the involved multiple theta functions. Using Stanley's Lemma on the fundamental parallelepiped, we show that a multiple theta function can be determined by a finite number of its coefficients. Thus such an identity can be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, including Riemann's addition formula and the extended Riemann identity. AMS Classification: 05E45, 14K25 Keywords: theta function, multiple theta function, contiguous relation, Jacobi's triple prod- uct identity, addition formula, Stanley's Lemma Download: pdf |